Continuity of a Function

I'm dealing right now with properties of a function and I have to prove if a given function is injective, surjective or bijective. I prove injectivity with the formula $x_1 = x_2 \Rightarrow f(x_1) = f(x_2)$ or $x_1 \neq x_2 \Rightarrow f(x_1) \neq f(x_2)$. For bijectivity I see if the function have an inverse function then it is automatically bijective. The problem is however with surjectivity. I know that if a function is continues at all points then it is surjective. I also know that using the formula $\lim_{x \to a} f(x)= f(a)$ one can prove continuity of ONE point. My problem is how to prove the continuity of a function at ALL points.

As I'm new to this subject in particular and to Analysis in general that would be great if you could explain it in details or even better with an example.

• I would prefer to you to go and study better the concept of surjectivity and bijectivity. If a function is bijective, it is automatically surjective as well! But surjectivity can't be seen from continuity of a function! You need to see if from an element of a range you have a corresponding element of domain. If you prove injectivity and surjectivity, than bijectivity is already proven!
– Emo
May 25, 2014 at 10:53

The implication $x_1=x_2\Rightarrow f(x_1)=f(x_2)$ does not mean that you function is injective but that it is well defined. Furthermore, continuity does not imply surjectivity, as you can see on the function $f:\Bbb{R}\rightarrow\Bbb{R}\cup\{\mbox{banana}\}$, $x\mapsto x$, which is clearly continuous, but not surjective.